SVD

Overview

The SVD function computes the Singular Value Decomposition of a given matrix, factorizing it into two unitary matrices and a diagonal matrix of singular values. This is a fundamental operation in linear algebra, used in applications such as signal processing, statistics, and machine learning for dimensionality reduction, pseudoinverse computation, and more. The decomposition is defined as:

A = U \cdot S \cdot V^H

where $A$ is the input matrix, $U$ and $V^H$ are unitary matrices, and $S$ is a diagonal matrix of singular values. For more details, see the scipy.linalg.svd documentation .

This example function is provided as-is without any representation of accuracy.

Usage

To use the function in Excel:


=SVD(matrix, [full_matrices], [compute_uv])

matrix (2D list, required): The matrix to decompose. Must be at least 2x2.
full_matrices (bool, optional, default=True): If True, U and Vh have full shapes; if False, reduced shapes.
compute_uv (bool, optional, default=True): If True, returns U, S, Vh; if False, returns only S.
overwrite_a (bool, optional, default=False): If True, allows overwriting the input matrix to improve performance.
check_finite (bool, optional, default=True): If True, checks that the input matrix contains only finite numbers.
lapack_driver (str, optional, default=‘gesdd’): LAPACK driver to use (‘gesdd’ or ‘gesvd’).
return_type (str, optional, default=‘u’): Controls the output. ‘u’, ‘s’, or ‘vh’ returns only the specified matrix as a 2D list. If ‘s’, singular values are returned as a 2D row vector. Only one matrix can be returned per call.

The function returns a 2D list: either U, S (as a diagonal matrix or row), or Vh, depending on return_type.

Examples

Example 1: U matrix of a 2x2 Matrix

In Excel:


=SVD({1,2;3,4}, TRUE, TRUE, FALSE, TRUE, "gesdd", "u")

Expected output:

U₁	U₂
-0.4046	-0.9145
-0.9145	0.4046

Example 2: S (singular values) of a 3x2 Matrix

In Excel:


=SVD({1,0;0,1;1,1}, FALSE, TRUE, TRUE, FALSE, "gesdd", "s")

Expected output:

S₁	S₂
1.7321	1.0

Example 3: Vh matrix of a 2x2 Matrix, Custom LAPACK Driver

In Excel:


=SVD({1,2;3,4}, TRUE, TRUE, FALSE, TRUE, "gesvd", "vh")

Expected output:

Vh₁	Vh₂
-0.576	-0.8174
0.8174	-0.576

Example 4: S (singular values) of a 2x3 Matrix

In Excel:


=SVD({1,2,3;4,5,6}, TRUE, TRUE, FALSE, TRUE, "gesdd", "s")

Expected output:

S₁	S₂
9.508	0.7729

Python Code


import numpy as np
from scipy.linalg import svd as scipy_svd
 
def svd(matrix, full_matrices=True, compute_uv=True, overwrite_a=False, check_finite=True, lapack_driver='gesdd', return_type='u'):
    """
    Compute the Singular Value Decomposition (SVD) of a matrix.
 
    Args:
        matrix: 2D list representing the matrix to decompose.
        full_matrices: If True, U and Vh have full shapes; if False, reduced shapes (default: True).
        compute_uv: If True, returns U, S, Vh; if False, returns only S (default: True).
        overwrite_a: If True, allows overwriting the input matrix to improve performance (default: False).
        check_finite: If True, checks that the input matrix contains only finite numbers (default: True).
        lapack_driver: LAPACK driver to use ('gesdd' or 'gesvd') (default: 'gesdd').
        return_type: 'u' (default), 's', or 'vh' to return only the specified matrix as a 2D list.
 
    Returns:
        2D list: The specified matrix (U, S, or Vh) as a 2D list. If 's', singular values are returned as a 2D row vector.
        Returns an error message (str or list[list[str]]) if input is invalid.
 
    This example function is provided as-is without any representation of accuracy.
    """
    # Validate input
    if not isinstance(matrix, list) or len(matrix) < 2 or not isinstance(matrix[0], list):
        return [["Invalid input: matrix must be a 2D list with at least 2 rows."]]
    try:
        arr = np.array(matrix, dtype=float)
    except Exception:
        return [["Invalid input: matrix must contain numeric values."]]
    try:
        if compute_uv:
            U, s, Vh = scipy_svd(arr, full_matrices=full_matrices, compute_uv=True, overwrite_a=overwrite_a, check_finite=check_finite, lapack_driver=lapack_driver)
            S = np.zeros_like(arr, dtype=float)
            for i in range(min(arr.shape)):
                S[i, i] = s[i]
            U = np.round(U, 4).tolist()
            S = np.round(S, 4).tolist()
            Vh = np.round(Vh, 4).tolist()
            if return_type == 'u':
                return U
            elif return_type == 's':
                return [s.round(4).tolist()]
            elif return_type == 'vh':
                return Vh
            else:
                return [["Invalid return_type"]]
        else:
            s = scipy_svd(arr, compute_uv=False, overwrite_a=overwrite_a, check_finite=check_finite, lapack_driver=lapack_driver)
            if return_type == 's':
                return [np.round(s, 4).tolist()]
            else:
                return [["Invalid return_type for compute_uv=False"]]
    except Exception as e:
        return [[f"SVD error: {e}"]]

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SVD

Overview

Usage

Examples

Python Code

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